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Background  Organisms that need to perform multiple tasks face a fundamental tradeoff: no design can be optimal at all tasks at once. Recent theory based on Pareto optimality showed that such tradeoffs lead to a highly defined range of phenotypes, which lie in low-dimensional polyhedra in the space of traits. The vertices of these polyhedra are called archetypes- the phenotypes that are optimal at a single task. To rigorously test this theory requires measurements of thousands of species over hundreds of millions of years of evolution. Ammonoid fossil shells provide an excellent model system for this purpose. Ammonoids have a well-defined geometry that can be parameterized using three dimensionless features of their logarithmic-spiral-shaped shells. Their evolutionary history includes repeated mass extinctions.    Results  We find that ammonoids fill out a pyramid in morphospace, suggesting five specific tasks - one for each vertex of the pyramid. After mass extinctions, surviving species evolve to refill essentially the same pyramid, suggesting that the tasks are unchanging. We infer putative tasks for each archetype, related to economy of shell material, rapid shell growth, hydrodynamics and compactness.    Conclusions  These results support Pareto optimality theory as an approach to study evolutionary tradeoffs, and demonstrate how this approach can be used to infer the putative tasks that may shape the natural selection of phenotypes.    Electronic supplementary material  The online version of this article (doi:10.1186/s12918-015-0149-z) contains supplementary material, which is available to authorized users.

Evolutionary tradeoffs, Pareto optimality and the morphology of ammonite shells